Algebra Examples

$\frac{x - 1}{x^{2} + x - 6} - \frac{x - 2}{x^{2} + 4 x + 3}$

Step 1

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 1.2

Write the factored form using these integers.

$\frac{x - 1}{x^{2} + x - 6} - \frac{x - 2}{(x + 1) (x + 3)}$

Step 2

To write $\frac{x - 1}{x^{2} + x - 6}$ as a fraction with a common denominator, multiply by $\frac{(x + 1) (x + 3)}{(x + 1) (x + 3)}$ .

$\frac{x - 1}{x^{2} + x - 6} \cdot \frac{(x + 1) (x + 3)}{(x + 1) (x + 3)} - \frac{x - 2}{(x + 1) (x + 3)}$

Step 3

To write $- \frac{x - 2}{(x + 1) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + x - 6}{x^{2} + x - 6}$ .

$\frac{x - 1}{x^{2} + x - 6} \cdot \frac{(x + 1) (x + 3)}{(x + 1) (x + 3)} - \frac{x - 2}{(x + 1) (x + 3)} \cdot \frac{x^{2} + x - 6}{x^{2} + x - 6}$

Step 4

Write each expression with a common denominator of $(x^{2} + x - 6) (x + 1) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x - 1}{x^{2} + x - 6}$ by $\frac{(x + 1) (x + 3)}{(x + 1) (x + 3)}$ .

$\frac{(x - 1) ((x + 1) (x + 3))}{(x^{2} + x - 6) ((x + 1) (x + 3))} - \frac{x - 2}{(x + 1) (x + 3)} \cdot \frac{x^{2} + x - 6}{x^{2} + x - 6}$

Step 4.2

Multiply $\frac{x - 2}{(x + 1) (x + 3)}$ by $\frac{x^{2} + x - 6}{x^{2} + x - 6}$ .

$\frac{(x - 1) ((x + 1) (x + 3))}{(x^{2} + x - 6) ((x + 1) (x + 3))} - \frac{(x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 4.3

Reorder the factors of $(x^{2} + x - 6) ((x + 1) (x + 3))$ .

$\frac{(x - 1) ((x + 1) (x + 3))}{(x + 1) (x + 3) (x^{2} + x - 6)} - \frac{(x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x - 1) ((x + 1) (x + 3)) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6

Rewrite $\frac{(x - 1) ((x + 1) (x + 3)) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$ in a factored form.

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Step 6.1

Expand $(x + 1) (x + 3)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{(x - 1) (x (x + 3) + 1 (x + 3)) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.1.2

Apply the distributive property.

$\frac{(x - 1) (x \cdot x + x \cdot 3 + 1 (x + 3)) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.1.3

Apply the distributive property.

$\frac{(x - 1) (x \cdot x + x \cdot 3 + 1 x + 1 \cdot 3) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

$\frac{(x - 1) (x^{2} + x \cdot 3 + 1 x + 1 \cdot 3) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.2.1.2

Move $3$ to the left of $x$ .

$\frac{(x - 1) (x^{2} + 3 x + 1 x + 1 \cdot 3) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.2.1.3

Multiply $x$ by $1$ .

$\frac{(x - 1) (x^{2} + 3 x + x + 1 \cdot 3) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.2.1.4

Multiply $3$ by $1$ .

$\frac{(x - 1) (x^{2} + 3 x + x + 3) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.2.2

Add $3 x$ and $x$ .

$\frac{(x - 1) (x^{2} + 4 x + 3) - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.3

Expand $(x - 1) (x^{2} + 4 x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x \cdot x^{2} + x (4 x) + x \cdot 3 - 1 x^{2} - 1 (4 x) - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4

Simplify each term.

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Step 6.4.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.4.1.1

Multiply $x$ by $x^{2}$ .

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Step 6.4.1.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{2} + x (4 x) + x \cdot 3 - 1 x^{2} - 1 (4 x) - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 2} + x (4 x) + x \cdot 3 - 1 x^{2} - 1 (4 x) - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4.1.2

Add $1$ and $2$ .

$\frac{x^{3} + x (4 x) + x \cdot 3 - 1 x^{2} - 1 (4 x) - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4.2

Rewrite using the commutative property of multiplication.

$\frac{x^{3} + 4 x \cdot x + x \cdot 3 - 1 x^{2} - 1 (4 x) - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4.3

Multiply $x$ by $x$ by adding the exponents.

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Step 6.4.3.1

Move $x$ .

$\frac{x^{3} + 4 (x \cdot x) + x \cdot 3 - 1 x^{2} - 1 (4 x) - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4.3.2

Multiply $x$ by $x$ .

$\frac{x^{3} + 4 x^{2} + x \cdot 3 - 1 x^{2} - 1 (4 x) - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4.4

Move $3$ to the left of $x$ .

$\frac{x^{3} + 4 x^{2} + 3 x - 1 x^{2} - 1 (4 x) - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4.5

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{x^{3} + 4 x^{2} + 3 x - x^{2} - 1 (4 x) - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4.6

Multiply $4$ by $- 1$ .

$\frac{x^{3} + 4 x^{2} + 3 x - x^{2} - 4 x - 1 \cdot 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.4.7

Multiply $- 1$ by $3$ .

$\frac{x^{3} + 4 x^{2} + 3 x - x^{2} - 4 x - 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.5

Subtract $x^{2}$ from $4 x^{2}$ .

$\frac{x^{3} + 3 x^{2} + 3 x - 4 x - 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.6

Subtract $4 x$ from $3 x$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - (x - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.7

Apply the distributive property.

$\frac{x^{3} + 3 x^{2} - x - 3 + (- x - - 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.8

Multiply $- 1$ by $- 2$ .

$\frac{x^{3} + 3 x^{2} - x - 3 + (- x + 2) (x^{2} + x - 6)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.9

Expand $(- x + 2) (x^{2} + x - 6)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{3} + 3 x^{2} - x - 3 - x \cdot x^{2} - x \cdot x - x \cdot - 6 + 2 x^{2} + 2 x + 2 \cdot - 6}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.10

Simplify each term.

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Step 6.10.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.10.1.1

Move $x^{2}$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - (x^{2} x) - x \cdot x - x \cdot - 6 + 2 x^{2} + 2 x + 2 \cdot - 6}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.10.1.2

Multiply $x^{2}$ by $x$ .

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Step 6.10.1.2.1

Raise $x$ to the power of $1$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - (x^{2} x^{1}) - x \cdot x - x \cdot - 6 + 2 x^{2} + 2 x + 2 \cdot - 6}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.10.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{3} + 3 x^{2} - x - 3 - x^{2 + 1} - x \cdot x - x \cdot - 6 + 2 x^{2} + 2 x + 2 \cdot - 6}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.10.1.3

Add $2$ and $1$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - x^{3} - x \cdot x - x \cdot - 6 + 2 x^{2} + 2 x + 2 \cdot - 6}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.10.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.10.2.1

Move $x$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - x^{3} - (x \cdot x) - x \cdot - 6 + 2 x^{2} + 2 x + 2 \cdot - 6}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.10.2.2

Multiply $x$ by $x$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - x^{3} - x^{2} - x \cdot - 6 + 2 x^{2} + 2 x + 2 \cdot - 6}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.10.3

Multiply $- 6$ by $- 1$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - x^{3} - x^{2} + 6 x + 2 x^{2} + 2 x + 2 \cdot - 6}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.10.4

Multiply $2$ by $- 6$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - x^{3} - x^{2} + 6 x + 2 x^{2} + 2 x - 12}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.11

Add $- x^{2}$ and $2 x^{2}$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - x^{3} + x^{2} + 6 x + 2 x - 12}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.12

Add $6 x$ and $2 x$ .

$\frac{x^{3} + 3 x^{2} - x - 3 - x^{3} + x^{2} + 8 x - 12}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.13

Subtract $x^{3}$ from $x^{3}$ .

$\frac{3 x^{2} - x - 3 + 0 + x^{2} + 8 x - 12}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.14

Add $3 x^{2}$ and $0$ .

$\frac{3 x^{2} - x - 3 + x^{2} + 8 x - 12}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.15

Add $3 x^{2}$ and $x^{2}$ .

$\frac{4 x^{2} - x - 3 + 8 x - 12}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.16

Add $- x$ and $8 x$ .

$\frac{4 x^{2} + 7 x - 3 - 12}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.17

Subtract $12$ from $- 3$ .

$\frac{4 x^{2} + 7 x - 15}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.18

Factor by grouping.

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Step 6.18.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 15 = - 60$ and whose sum is $b = 7$ .

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Step 6.18.1.1

Factor $7$ out of $7 x$ .

$\frac{4 x^{2} + 7 (x) - 15}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.18.1.2

Rewrite $7$ as $- 5$ plus $12$

$\frac{4 x^{2} + (- 5 + 12) x - 15}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.18.1.3

Apply the distributive property.

$\frac{4 x^{2} - 5 x + 12 x - 15}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.18.2

Factor out the greatest common factor from each group.

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Step 6.18.2.1

Group the first two terms and the last two terms.

$\frac{(4 x^{2} - 5 x) + 12 x - 15}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.18.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (4 x - 5) + 3 (4 x - 5)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.18.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 5$ .

$\frac{(4 x - 5) (x + 3)}{(x + 1) (x + 3) (x^{2} + x - 6)}$

Step 6.19

Factor.

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Step 6.19.1

Factor $x^{2} + x - 6$ using the AC method.

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Step 6.19.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 6.19.1.2

Write the factored form using these integers.

$\frac{(4 x - 5) (x + 3)}{(x + 1) (x + 3) ((x - 2) (x + 3))}$

Step 6.19.2

Remove unnecessary parentheses.

$\frac{(4 x - 5) (x + 3)}{(x + 1) (x + 3) (x - 2) (x + 3)}$

Step 6.20

Combine exponents.

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Step 6.20.1

Raise $x + 3$ to the power of $1$ .

$\frac{(4 x - 5) (x + 3)}{(x + 1) ({(x + 3)}^{1} (x + 3)) (x - 2)}$

Step 6.20.2

Raise $x + 3$ to the power of $1$ .

$\frac{(4 x - 5) (x + 3)}{(x + 1) ({(x + 3)}^{1} {(x + 3)}^{1}) (x - 2)}$

Step 6.20.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(4 x - 5) (x + 3)}{(x + 1) {(x + 3)}^{1 + 1} (x - 2)}$

Step 6.20.4

Add $1$ and $1$ .

$\frac{(4 x - 5) (x + 3)}{(x + 1) {(x + 3)}^{2} (x - 2)}$

Step 6.21

Reduce the expression $\frac{(4 x - 5) (x + 3)}{(x + 1) {(x + 3)}^{2} (x - 2)}$ by cancelling the common factors.

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Step 6.21.1

Factor $x + 3$ out of $(4 x - 5) (x + 3)$ .

$\frac{(x + 3) (4 x - 5)}{(x + 1) {(x + 3)}^{2} (x - 2)}$

Step 6.21.2

Factor $x + 3$ out of $(x + 1) {(x + 3)}^{2} (x - 2)$ .

$\frac{(x + 3) (4 x - 5)}{(x + 3) (((x + 1) (x + 3)) (x - 2))}$

Step 6.21.3

Cancel the common factor.

$\frac{(x + 3) (4 x - 5)}{(x + 3) (((x + 1) (x + 3)) (x - 2))}$

Step 6.21.4

Rewrite the expression.

$\frac{4 x - 5}{((x + 1) (x + 3)) (x - 2)}$